Unformatted text preview: 1 + 0 = 0+ = 1 0 + 0 = 0
Row 2 00 = 0
0- 1 = 1 0
1-1 = 1
= -0 =
Figure 6.4. Illustrating the principle of duality in Boolean algebra.
The implication of this principle is that any theorem in Boolean algebra has its dual
obtainable by interchanging '+' with '•' and '0' with ' 1'. Thus if a particular theorem is
proved, its dual theorem automatically holds and need not be proved separately.
Theorems of Boolean Algebra
Some of the important theorems of Boolean algebra are stated below along with their
proof. Theorem 1 (Idempotent law)
(a) x + x = x
(b) x • x = x
Proofof(a) Proof of (b) L.H.S. L.H.S. =X+X =x•x = (x + x) • 1 by postulate 2(b) =x-x+0 by postulate 2(a) by postulate 6(a) =X■X+X• X by postulate 6(b) =X+X• X by postulate 5(b) = x • (x + x ) by postulate 5(a) =x+0 by postulate 6(b) =X•1 by postulate 6(a) =x by postulate 2(a) =x by postulate 2(b) = (x + x) • (x + = R.H.S. X) = R.H.S. Note that Theorem l(b). is the dual of Theorem l(a) and that each step of the proof in part
(b) is the dual of part (a). Any dual theorem can be similarly derived from the proof of its
corresponding pair. Hence from now onwards, the proof of part (a) only will be given.
Interested readers can apply the principle of duality to the various steps of the proof of
part (a) to obtain the proof of part (b) for any theorem.
Theorem 3 (Absorption law)
(a) x + x . y = x
(b) x • (x + y) = x
Proof of (a) Proof by the Method of Perfect Induction
The theorems of Boolean algebra can also be proved by means of truth tables. In a truth
table, both sides of the relation are checked to yield identical results for all possible
combinations of variables involved. In principle, it is possible to enumerate all possible
combinations of the variables involved because Boolean algebra deals with variables that
can have only two values. This method of proving theorems is called exhaustive
enumeration or perfect induction.
For example, Figure 6.5 is a truth table for proving Theorem 3(a) by the method of
perfect induction. S...
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This document was uploaded on 04/07/2014.
- Spring '14